Generalized Binary Search For Split-Neighborly Problems
This work addresses the efficiency of hypothesis testing algorithms for researchers in machine learning and optimization, though it appears incremental as it extends an existing condition to a broader class of problems.
The paper tackles the problem of achieving optimal query cost in sequential hypothesis testing by introducing a weaker condition called split-neighborly, which generalizes the k-neighborly condition. For four problems that are not k-neighborly, they prove split-neighborly holds, resulting in an optimal O(log n) worst-case query cost.
In sequential hypothesis testing, Generalized Binary Search (GBS) greedily chooses the test with the highest information gain at each step. It is known that GBS obtains the gold standard query cost of $O(\log n)$ for problems satisfying the $k$-neighborly condition, which requires any two tests to be connected by a sequence of tests where neighboring tests disagree on at most $k$ hypotheses. In this paper, we introduce a weaker condition, split-neighborly, which requires that for the set of hypotheses two neighbors disagree on, any subset is splittable by some test. For four problems that are not $k$-neighborly for any constant $k$, we prove that they are split-neighborly, which allows us to obtain the optimal $O(\log n)$ worst-case query cost.